Eisenstein series

Relation to modular invariants

The modular invariants ${\displaystyle g_2}$ and ${\displaystyle g_3}$ of an elliptic curve are given by the first two Eisenstein series:

${\displaystyle \begin{array}{cc}{g}_2& =60G_4\\ g_3& =140G_6.\end{array}}$

The article on modular invariants provides expressions for these two functions in terms of theta functions.

Recurrence relation

Any holomorphic modular form for the modular group can be written as a polynomial in ${\displaystyle G_4}$ and ${\displaystyle G_6}$. Specifically, the higher order ${\displaystyle G_{2k}}$ can be written in terms of ${\displaystyle G_4}$ and ${\displaystyle G_6}$ through a recurrence relation. Let ${\displaystyle d_k=(2k+3)!G_{2k+4}}$. Then the ${\displaystyle d_k}$ satisfy the relation

${\displaystyle \sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})d_k{d}_{n-k}=\frac{2n+9}{3n+6}{d}_{n+2}}$

for all ${\displaystyle n\ge 0}$.

The ${\displaystyle d_k}$ occur in the series expansion for Weierstrass's elliptic functions:

${\displaystyle \begin{array}{cc}\mathrm{\wp }(z)& =\frac{1}{z^2}+z^2\sum _{k=0}^{\mathrm{\infty }}\frac{d_k{z}^{2k}}{k!}\\ & =\frac{1}{z^2}+\sum _{k=1}^{\mathrm{\infty }}(2k+1)G_{2k+2}{z}^{2k}.\end{array}}$

Fourier series

Let ${\displaystyle q=e^{2\pi i\tau }}$. Then the Fourier series of the Eisenstein series is

${\displaystyle G_{2k}(\tau )=2\zeta (2k)(1+c_{2k}\sum _{n=1}^{\mathrm{\infty }}{\sigma }_{2k-1}(n)q^n).}$

The coefficients ${\displaystyle c_{2k}}$ are given by

${\displaystyle c_{2k}=\frac{(2\pi i{)}^{2k}}{(2k-1)!\zeta (2k)}=\frac{-4k}{B_{2k}}=\frac{2}{\zeta (1-2k)},}$

where ${\displaystyle B_n}$ are the Bernoulli numbers, ${\displaystyle \zeta }$ is the Riemann zeta function and ${\displaystyle {\sigma }_p(n)}$ is the divisor sum function, i.e. the sum of the ${\displaystyle p}$-th powers of the divisors of ${\displaystyle n}$. In particular, one has

${\displaystyle G_4(\tau )=\frac{{\pi }^4}{45}(1+240\sum _{n=1}^{\mathrm{\infty }}{\sigma }_3(n)q^n)}$

and

${\displaystyle G_6(\tau )=\frac{2{\pi }^6}{945}(1-504\sum _{n=1}^{\mathrm{\infty }}{\sigma }_5(n)q^n).}$

The summation over ${\displaystyle q}$ can be resummed as a Lambert series; that is, one has

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{q}^n{\sigma }_a(n)=\sum _{n=1}^{\mathrm{\infty }}\frac{n^a{q}^n}{1-q^n}}$

for arbitrary complex ${\displaystyle |q|<1}$ and ${\displaystyle a}$. When working with the q-expansion of the Eisenstein series, this alternate notation is frequently introduced:

${\displaystyle \begin{array}{cc}{E}_{2k}(\tau )& =\frac{G_{2k}(\tau )}{2\zeta (2k)}\\ & =1+\frac{2}{\zeta (1-2k)}\sum _{n=1}^{\mathrm{\infty }}\frac{n^{2k-1}{q}^n}{1-q^n}\\ & =1-\frac{4k}{B_{2k}}\sum _{n=1}^{\mathrm{\infty }}{\sigma }_{2k-1}(n)q^n\\ & =1-\frac{4k}{B_{2k}}\sum _{d,n\ge 1}{n}^{2k-1}{q}^{nd}.\end{array}}$

Identities involving Eisenstein series

Theta functions

Given ${\displaystyle q=e^{2\pi i\tau }}$, let

${\displaystyle \begin{array}{cc}{E}_4(\tau )& =1+240\sum _{n=1}^{\mathrm{\infty }}\frac{n^3{q}^n}{1-q^n},\\ E_6(\tau )& =1-504\sum _{n=1}^{\mathrm{\infty }}\frac{n^5{q}^n}{1-q^n},\\ E_8(\tau )& =1+480\sum _{n=1}^{\mathrm{\infty }}\frac{n^7{q}^n}{1-q^n},\end{array}}$

and define the Jacobi theta functions (which normally use the nome ${\displaystyle e^{\pi i\tau }}$)

${\displaystyle \begin{array}{cc}a& ={\theta }_2\left(0;e^{\pi i\tau }\right)={\vartheta }_{10}(0;\tau ),\\ b& ={\theta }_3\left(0;e^{\pi i\tau }\right)={\vartheta }_{00}(0;\tau ),\\ c& ={\theta }_4\left(0;e^{\pi i\tau }\right)={\vartheta }_{01}(0;\tau )\end{array},}$

where ${\displaystyle {\theta }_m}$ and ${\displaystyle {\vartheta }_{ij}}$ are alternative notations. Then we have the symmetric relations

${\displaystyle \begin{array}{cc}{E}_4(\tau )& =\frac{1}{2}\left(a^8+b^8+c^8\right),\\ E_6(\tau )& =\frac{1}{2}\sqrt{\frac{{\left(a^8+b^8+c^8\right)}^3-54(abc{)}^8}{2}},\\ E_8(\tau )& =\frac{1}{2}\left(a^{16}+b^{16}+c^{16}\right)=a^8{b}^8+a^8{c}^8+b^8{c}^8.\end{array}}$

Basic algebra immediately implies

${\displaystyle E_4^3-E_6^2=\frac{27}{4}(abc{)}^8,}$

an expression related to the modular discriminant

${\displaystyle \mathrm{\Delta }=g_2^3-27g_3^2=(2\pi {)}^{12}(\frac{abc}{2}{)}^8}$

The third symmetric relation is a consequence of ${\displaystyle E_8=E_4^2}$ and ${\displaystyle a^4-b^4+c^4=0}$.

Products of Eisenstein series

Eisenstein series form the most explicit examples of modular forms for the full modular group ${\displaystyle \mathrm{SL}(2,\mathbb{Z})}$. Since the space of modular forms of weight ${\displaystyle 2k}$ has dimension 1 for ${\displaystyle 2k=4,6,8,10,14}$, different products of Eisenstein series having those weights have to be equal up to a scalar multiple. In fact, we obtain the identities

${\displaystyle E_4^2=E_8,E_4{E}_6=E_{10},E_4{E}_{10}=E_{14},E_6{E}_8=E_{14}.}$

Using the ${\displaystyle q}$-expansions of the Eisenstein series given above, they may be restated as identities involving the sums of powers of divisors:

${\displaystyle (1+240\sum _{n=1}^{\mathrm{\infty }}{\sigma }_3(n)q^n{)}^2=1+480\sum _{n=1}^{\mathrm{\infty }}{\sigma }_7(n)q^n,}$

hence

${\displaystyle {\sigma }_7(n)={\sigma }_3(n)+120\sum _{m=1}^{n-1}{\sigma }_3(m){\sigma }_3(n-m),}$

and similarly for the others. The theta function of an eight-dimensional even unimodular lattice ${\displaystyle \mathrm{\Gamma }}$ is a modular form of weight 4 for the full modular group, which gives the following identities:

${\displaystyle {\theta }_{\mathrm{\Gamma }}(\tau )=1+\sum _{n=1}^{\mathrm{\infty }}{r}_{\mathrm{\Gamma }}(2n)q^n=E_4(\tau ),r_{\mathrm{\Gamma }}(n)=240{\sigma }_3(n)}$

for the number ${\displaystyle r_{\mathrm{\Gamma }}(n)}$ of vectors of the squared length ${\displaystyle 2n}$ in the root lattice of the type E₈.

Similar techniques involving holomorphic Eisenstein series twisted by a Dirichlet character produce formulas for the number of representations of a positive integer ${\displaystyle n}$' as a sum of two, four, or eight squares in terms of the divisors of ${\displaystyle n}$.

Using the above recurrence relation, all higher ${\displaystyle E_{2k}}$ can be expressed as polynomials in ${\displaystyle E_4}$ and ${\displaystyle E_6}$. For example,

${\displaystyle \begin{array}{cc}{E}_8& =E_4^2\\ E_{10}& =E_4\cdot E_6\\ 691\cdot E_{12}& =441\cdot E_4^3+250\cdot E_6^2\\ E_{14}& =E_4^2\cdot E_6\\ 3617\cdot E_{16}& =1617\cdot E_4^4+2000\cdot E_4\cdot E_6^2\\ 43867\cdot E_{18}& =38367\cdot E_4^3\cdot E_6+5500\cdot E_6^3\\ 174611\cdot E_{20}& =53361\cdot E_4^5+121250\cdot E_4^2\cdot E_6^2\\ 77683\cdot E_{22}& =57183\cdot E_4^4\cdot E_6+20500\cdot E_4\cdot E_6^3\\ 236364091\cdot E_{24}& =49679091\cdot E_4^6+176400000\cdot E_4^3\cdot E_6^2+10285000\cdot E_6^4\end{array}}$

Many relationships between products of Eisenstein series can be written in an elegant way using Hankel determinants, such as Garvan's identity

${\displaystyle {\left(\frac{\mathrm{\Delta }}{(2\pi {)}^{12}}\right)}^2=-\frac{691}{{1728}^2\cdot 250}\left|\begin{array}{ccc}{E}_4& E_6& E_8\\ E_6& E_8& E_{10}\\ E_8& E_{10}& E_{12}\end{array}\right|,}$

where

${\displaystyle \mathrm{\Delta }=(2\pi {)}^{12}\frac{E_4^3-E_6^2}{1728}}$

is the modular discriminant.

Ramanujan identities

Srinivasa Ramanujan gave several interesting identities between the first few Eisenstein series involving differentiation. Let

${\displaystyle \begin{array}{cc}L(q)& =1-24\sum _{n=1}^{\mathrm{\infty }}\frac{n{q}^n}{1-q^n}=E_2(\tau ),\\ M(q)& =1+240\sum _{n=1}^{\mathrm{\infty }}\frac{n^3{q}^n}{1-q^n}=E_4(\tau ),\\ N(q)& =1-504\sum _{n=1}^{\mathrm{\infty }}\frac{n^5{q}^n}{1-q^n}=E_6(\tau ),\end{array}}$

then

${\displaystyle \begin{array}{cc}q\frac{dL}{dq}& =\frac{L^2-M}{12},\\ q\frac{dM}{dq}& =\frac{LM-N}{3},\\ q\frac{dN}{dq}& =\frac{LN-M^2}{2}.\end{array}}$

These identities, like the identities between the series, yield arithmetical convolution identities involving the sum-of-divisor function. Following Ramanujan, to put these identities in the simplest form it is necessary to extend the domain of ${\displaystyle {\sigma }_p(n)}$ to include zero, by setting

${\displaystyle {\sigma }_p(0)=\frac{\zeta (-p)}{2}.}$

Then, for example,

${\displaystyle \sum _{k=0}^n\sigma (k)\sigma (n-k)=\frac{5}{12}{\sigma }_3(n)-\frac{n\sigma (n)}{2}.}$

Other identities of this type, but not directly related to the preceding relations between ${\displaystyle L}$, ${\displaystyle M}$ and ${\displaystyle N}$ functions, have been proved, as for example

${\displaystyle \begin{array}{cc}& \sum _{k=0}^n{\sigma }_3(k){\sigma }_3(n-k)=\frac{1}{120}{\sigma }_7(n),\\ & \sum _{k=0}^n\sigma (2k+1){\sigma }_3(n-k)=\frac{1}{240}{\sigma }_5(2n+1),\\ & \sum _{k=0}^n\sigma (3k+1)\sigma (3n-3k+1)=\frac{1}{9}{\sigma }_3(3n+2).\end{array}}$

Generalizations

Automorphic forms generalize the idea of modular forms for general semisimple Lie groups, where the modular group is replaced by an arithmetic group. Robert Langlands generalised the theory of Eisenstein series to this setting.

When the Lie group is of type A₁ the theory resembles the classical case. For example, Hilbert modular forms are well-studied.